Distributed Generalized Distributionally Robust Equilibrium Seeking for Dynamical Games Under Unknown Time-Varying Interference

Longcheng Liu; Shuai Liu; Yiguang Hong; Lihua Xie; Guangchen Wang

doi:10.1109/JAS.2025.125462

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2026 > In Press, Accepted Manuscript

L. Liu, S. Liu, Y. Hong, L. Xie, and G. Wang, “Distributed generalized distributionally robust equilibrium seeking for dynamical games under unknown time-varying interference,” IEEE/CAA J. Autom. Sinica, early access, 2026. doi: 10.1109/JAS.2025.125462

Citation:

L. Liu, S. Liu, Y. Hong, L. Xie, and G. Wang, “Distributed generalized distributionally robust equilibrium seeking for dynamical games under unknown time-varying interference,” IEEE/CAA J. Autom. Sinica, early access, 2026. doi: 10.1109/JAS.2025.125462

Citation:

L. Liu, S. Liu, Y. Hong, L. Xie, and G. Wang, “Distributed generalized distributionally robust equilibrium seeking for dynamical games under unknown time-varying interference,” IEEE/CAA J. Autom. Sinica, early access, 2026. doi: 10.1109/JAS.2025.125462

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Distributed Generalized Distributionally Robust Equilibrium Seeking for Dynamical Games Under Unknown Time-Varying Interference

doi: 10.1109/JAS.2025.125462

Funds: This work was supported in part by the National Natural Science Foundation of China (62373226, 62133008)

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Author Bio:
Longcheng Liu received the B.E. degree and M.E degree in control theory and engineering from Shandong University in 2021 and 2024, respectively. He is currently working toward the Ph.D. degree in control theory and control engineering from Shandong University. His research interests include distributed optimization, game theory, distributionally robust optimization, and multi-agent system

Shuai Liu (Member, IEEE) received the B.E. and M.E. degrees in control theory and engineering from Shandong University in 2004 and 2007, respectively, and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 2012. From 2011 to 2017, he was a Senior Research Fellow with Berkeley Education Alliance, Singapore. Since 2017, he has been with the School of Control Science and Engineering, Shandong University. He serves as an Associate Editor for several high-impact journals in the control field, including IEEE/CAA Journal of Automatica Sinica, IEEE Transactions on Cybernetics, ISA Transactions and Science China Information Sciences. His research interests include distributed control, estimation and optimization, integrated energy system, and learning algorithms

Yiguang Hong (Fellow, IEEE) received the B.S. and M.S. degrees from Peking University in 1987 and 1990, respectively, and the Ph.D. degree from the Chinese Academy of Sciences (CAS) in 1993. He has been a Professor at the Shanghai Institute of Intelligent Science and Technology, Tongji University, since October 2020, and was a Professor at the Academy of Mathematics and Systems Science, CAS. His current research interests include nonlinear control, multi-agent systems, distributed optimization/game, machine learning, and social networks. Prof. Hong serves as Editor-in-Chief of Control Theory and Technology. He also serves or served as Associate Editors for many journals, including the IEEE Transactions on Automatic Control, IEEE Transactions on Control of Network Systems, and IEEE Control Systems Magazine. He is a recipient of the Guan Zhaozhi Award at the Chinese Control Conference, Young Author Prize of the IFAC World Congress, Young Scientist Award of CAS, the Youth Award for Science and Technology of China, and the National Natural Science Prize of China. He is also a Fellow of IEEE, a Fellow of Chinese Association for Artificial Intelligence, and a Fellow of Chinese Association of Automation (CAA). Additionally, he is the Chair of Technical Committee of Control Theory of CAA and was a board of Governor of IEEE Control Systems Society

Lihua Xie (Fellow, IEEE) received the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently President’s Chair and Director, Center for Advanced Robotics Technology Innovation. He has served as the Head of Division of Control and Instrumentation and Co-Director, Delta-NTU Corporate Lab for Cyber-Physical Systems. He held teaching appointments in the Department of Automatic Control, Nanjing University of Science and Technology from 1986 to 1989. Dr Xie’s research interests include robust control and estimation, networked control systems, multi-agent networks, and unmanned systems. He has published 10 books and numerous papers in the areas, and holds 25 patents/technical disclosures. He was listed as a highly cited researcher by Thomson Routers and Clarivate Analytics. He is an Editor-in-Chief for Unmanned Systems and has served as Editor for IET Book Series in Control and Associate Editor for a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control Systems Technology, IEEE Transactions on Network Control Systems, etc. He was an IEEE Distinguished Lecturer (January 2012 to December 2014) and the General Chair of the 62nd IEEE Conference on Decision and Control (CDC 2023). He is currently Vice-President of IEEE Control System Society. Dr. Xie is Fellow of Academy of Engineering Singapore, IEEE, IFAC, and CAA

Guangchen Wang (Senior Member, IEEE) received the B.S. degree in mathematics from Shandong Normal University in 2001, and the Ph.D. degree in probability theory and mathematical statistics from the School of Mathematics and System Sciences, Shandong University in 2007. From July 2007 to August 2010, he served as a Lecturer at the School of Mathematical Sciences, Shandong Normal University. He joined the School of Control Science and Engineering, Shandong University in September 2010, where he has been a Full Professor since September 2014. He is currently a Distinguished Young Scholar of NSF of China. His current research interests include stochastic control, optimal filtering, and mathematical finance
Corresponding author: Shuai Liu, e-mail: liushuai@sdu.edu.cn
Received Date: 2025-02-18
Accepted Date: 2025-04-02

Available Online: 2026-03-03

Abstract

Abstract

This paper investigates a distributed generalized Nash equilibrium-seeking problem in stochastic dynamical systems, focusing on two key challenges: 1) nonlinear coupled constraints and dynamics in player states, and 2) nonconvex objectives influenced by disturbances with unknown time-varying distributions. To address these challenges, a distributionally robust game framework with an exact penalty is proposed. We introduce a first-order equilibrium concept suitable for nonconvex-nonsmooth settings and ensure finite-sample guarantees. Furthermore, a distributed zeroth-order feedback algorithm is proposed to solve the problem. This algorithm utilizes gradient estimators for the objective functions and subgradient estimators for the exact penalty terms. We provide a detailed analysis of the relationship between communication errors and the dynamic energy of the system, along with an expected upper bound for the zeroth-order gradient estimation. Our findings indicate that the expectation of the time-accumulated regret grows at a sublinear rate. Furthermore, as the distribution stabilizes, we show that the empirical distribution converges with ${\boldsymbol{O(1)}}$ sampling complexity.
- Distributionally robust game,
- dynamical system,
- equilibrium seeking,
- non-convex non-smooth games,
- stochastic game

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References

[1]	J. Zhou, G. Wen, Y. Lv, T. Yang, and G. Chen, “Intra-independent distributed resource allocation game,” IEEE/CAA J. Autom. Sinica, vol. 12, no. 10, pp. 2150–2152, 2025.
[2]	A. Deligiannis, A. Panoui, S. Lambotharan, and J. A. Chambers, “Game-theoretic power allocation and the Nash equilibrium analysis for a multistatic MIMO radar network,” IEEE Trans. Signal Processing, vol. 65, no. 24, pp. 6397–6408, 2017. doi: 10.1109/TSP.2017.2755591
[3]	X. Cai, F. Xiao, and B. Wei, “Resilient Nash equilibrium seeking in multiagent games under false data injection attacks,” IEEE Trans. Systems, Man, and Cybernetics: Systems, vol. 53, no. 1, pp. 275–284, 2023. doi: 10.1109/TSMC.2022.3180006
[4]	G. Scutari, D. P. Palomar, and S. Barbarossa, “Optimal linear precoding strategies for wideband noncooperative systems based on game theory — Part I: Nash equilibria,” IEEE Trans. Signal Processing, vol. 56, no. 3, pp. 1230–1249, 2008. doi: 10.1109/TSP.2007.907807
[5]	J. Huang, X. Wang, Y. Wang, C. Shao, G. Chen, and P. Wang, “A game-theoretic approach for electric vehicle aggregators participating in phase balancing considering network topology,” IEEE Trans. Smart Grid, vol. 15, no. 1, pp. 743–756, 2024. doi: 10.1109/TSG.2023.3276242
[6]	M. Ye, Q.-L. Han, L. Ding, S. Xu, and G. Jia, “Distributed Nash equilibrium seeking strategies under quantized communication,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 1, pp. 103–112, 2024. doi: 10.1109/JAS.2022.105857
[7]	H. Xu, K. Lu, T. Wang, X. Yan, and Q. Zhu, “Online distributed seeking for first-order Nash equilibria of nonconvex noncooperative games with multiple clusters,” IEEE Trans. Circuits and Systems II: Express Briefs, vol. 70, no. 2, pp. 621–625, Feb. 2023.
[8]	G. Belgioioso, D. Liao-McPherson, M. Hudoba de Badyn, S. Bolognani, R. S. Smith, J. Lygeros, and F. Dörfler, “Online feedback equilibrium seeking,” IEEE Trans. Autom. Control, vol. 70, no. 1, pp. 203−218, Jan. 2025.
[9]	Z. He, S. Bolognani, J. He, F. Dörfler, and X. Guan, “Model-free nonlinear feedback optimization,” IEEE Trans. Autom. Control, vol. 69, no. 7, pp. 4554−4569, 2024.
[10]	M. Ye, “On resilience against cyber-physical uncertainties in distributed Nash equilibrium seeking strategies for heterogeneous games,” IEEE/CAA J. Autom. Sinica, vol. 12, no. 1, pp. 138–147, 2025. doi: 10.1109/JAS.2024.124983
[11]	F. Niu, X. Nian, Y. Chen, M. Lv, J. Huang, and B. Hao, “Distributed time-varying Nash equilibrium in resilient multi-objective formation control for cyber–physical systems,” J. the Franklin Institute, vol. 361, p. 106903, 2024. doi: 10.1016/j.jfranklin.2024.106903
[12]	L. Xue, J. Ye, Y. Wu, J. Liu, and D. C. Wunsch, “Prescribed-time Nash equilibrium seeking for pursuit-evasion game,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 6, pp. 1518–1520, 2024. doi: 10.1109/JAS.2023.124077
[13]	R. Yu, M. Meng, and L. Li, “Distributed online learning for leaderless multicluster games in dynamic environments,” IEEE Trans. Control of Network Systems, vol. 11, no. 3, pp. 1548–1561, 2024. doi: 10.1109/TCNS.2023.3339304
[14]	K. Lu and L. Wang, “Online distributed optimization with nonconvex objective functions via dynamic regrets,” IEEE Trans. Autom. Control, vol. 68, no. 11, pp. 6509–6524, Nov. 2023. doi: 10.1109/TAC.2023.3239432
[15]	N. Fournier and A. Guillin, “On the rate of convergence in Wasserstein distance of the empirical measure,” Probability Theory and Related Fields, vol. 162, pp. 707–738, 2015. doi: 10.1007/s00440-014-0583-7
[16]	K. Akkarajitsakul, E. Hossain, and D. Niyato, “Coalition-based cooperative packet delivery under uncertainty: A dynamic bayesian coalitional game,” IEEE Trans. Mobile Computing, vol. 12, no. 2, pp. 371–385, 2013. doi: 10.1109/TMC.2011.251
[17]	P. Yao, Z. Jiang, B. Yan, Q. Yang, and W. Wang, “Bayesian and stochastic game joint approach for cross-layer optimal defensive decision-making in industrial cyber-physical systems,” Information Sciences, vol. 662, p. 120216, 2024. doi: 10.1016/j.ins.2024.120216
[18]	M. Tian, Z. Dong, and X. Wang, “Analysis of false data injection attacks in power systems: A dynamic Bayesian game-theoretic approach,” ISA Transactions, vol. 115, pp. 108–123, 2021. doi: 10.1016/j.isatra.2021.01.011
[19]	S. Morris, “The common prior assumption in economic theory,” Economics and Philosophy, vol. 1995, no. 11, pp. 227–253, 1995.
[20]	M. Aghassi and D. Bertsimas, “Robust game theory,” Mathematical Programming, vol. 107, pp. 231–273, 2006. doi: 10.1007/s10107-005-0686-0
[21]	F. Farokhi, “Distributionally robust optimization with noisy data for discrete uncertainties using total variation distance,” IEEE Control Systems Letters, vol. 7, pp. 1494–1499, 2023. doi: 10.1109/LCSYS.2023.3271434
[22]	A. Cherukuri and J. Cortés, “Cooperative data-driven distributionally robust optimization,” IEEE Trans. Autom. Control, vol. 65, no. 10, pp. 4400–4407, 2020. doi: 10.1109/TAC.2019.2955031
[23]	F. Fabiani and B. Franci, “On distributionally robust generalized Nash games defined over the Wasserstein ball,” J. Optimization Theory and Applications, vol. 199, pp. 298–309, 2023. doi: 10.1007/s10957-023-02284-3
[24]	M. S. Ali and N. B. Mehta, “Modeling time-varying aggregate interference in cognitive radio systems, and application to primary exclusive zone design,” IEEE Trans. Wireless Communications, vol. 13, no. 1, pp. 429–439, 2014. doi: 10.1109/TWC.2013.113013.130762
[25]	B. Huang, Y. Liu, K. Kou, and W. Gui, “Multi-timescale distributed approach to generalized-Nash-equilibrium seeking in noncooperative nonconvex games,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 3, pp. 791–793, 2024. doi: 10.1109/JAS.2023.123909
[26]	M. Meng and X. Li, “On the linear convergence of distributed Nash equilibrium-seeking for multi-cluster games under partial-decision information,” Automatica, vol. 151, p. 110919, 2023. doi: 10.1016/j.automatica.2023.110919
[27]	Z. Deng, Y. Liu, and T. Chen, “Generalized Nash equilibrium-seeking algorithm design for distributed constrained noncooperative games with second-order players,” Automatica, vol. 141, p. 110317, 2022. doi: 10.1016/j.automatica.2022.110317
[28]	G. Scutari, F. Facchinei and L. Lampariello, “Parallel and distributed methods for constrained nonconvex optimization — Part I: Theory,” IEEE Trans. Signal Processing, vol. 65, no. 8, pp. 1929–1944, 2017. doi: 10.1109/TSP.2016.2637317
[29]	C. L. Su, “A sequential NCP algorithm for solving equilibrium problems with equilibrium constraints,” Dept. of Management Sci. & Eng., Stanford Univ., Stanford, CA, Tech. Rep., Oct. 2004.
[30]	C. Sun and G. Hu, “Continuous-time penalty methods for Nash equilibrium-seeking of a nonsmooth generalized noncooperative game,” IEEE Trans. Autom. Control, vol. 66, no. 10, pp. 4895–4902, 2021. doi: 10.1109/TAC.2020.3040377
[31]	A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, New York, NY, USA: Springer, 2009.
[32]	D. Bertsekas, “Nonlinear programming,” J. the Operational Research Society, vol. 48, pp. 334–334, 1997. doi: 10.1057/palgrave.jors.2600425
[33]	R. A. Poliquin and R. T. Rockafellar, “Prox-regular functions in variational analysis,” Trans. of the AMS — American Mathematical Society, vol. 348, pp. 1805–1838, 1996. doi: 10.1090/S0002-9947-96-01544-9
[34]	Y. Hong, J. Hu, and L. Gao, “Tracking control for multi-agent consensus with an active leader and variable topology,” Automatica, vol. 42, pp. 1177–1182, 2006. doi: 10.1016/j.automatica.2006.02.013
[35]	Y. Nesterov and V. Spokoiny, “Random gradient-free minimization of convex functions,” Foundations of Computational Mathematics, vol. 17, no. 2, pp. 527–566, 2017. doi: 10.1007/s10208-015-9296-2
[36]	P. Liu, Z. Fu, J. Cao, Y. Wei, J. Guo, and W. Huang, “A decentralized strategy for generalized Nash equilibrium with linear coupling constraints,” Mathematics and Computers in Simulation, vol. 171, pp. 221–232, 2020. doi: 10.1016/j.matcom.2019.06.004
[37]	W. He and Y. Wang, “Distributed optimal variational GNE seeking in merely monotone games,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 7, pp. 1621–1630, 2024. doi: 10.1109/JAS.2024.124284
[38]	S. Leyffer and T. Munson, “Solving multi-leader-common-follower games,” Optimization Methods Software, vol. 25, pp. 601–623, 2010. doi: 10.1080/10556780903448052
[39]	T. H. Nguyen and L. Xie, “Estimating odometry scale and UWB anchor location based on semidefinite programming optimization,” IEEE Robotics and Autom. Letters, vol. 7, no. 3, pp. 7359–7366, 2022. doi: 10.1109/LRA.2022.3182110
[40]	B. Zhou, Y. Lv, Y. Mao, J. Wang, S. Yu, and Q. Xuan, “The robustness of graph k-Shell structure under adversarial attacks,” IEEE Trans. Circuits and Systems II: Express Briefs, vol. 69, no. 3, pp. 1797–1801, Mar. 2022.

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Distributed Generalized Distributionally Robust Equilibrium Seeking for Dynamical Games Under Unknown Time-Varying Interference

doi: 10.1109/JAS.2025.125462

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